Theorem 2omotap 7521

Description: If there is at most one tight apartness on 2_o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2omotap	⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Proof of Theorem 2omotap

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2omotaplemst 7520	. . . . 5 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝑥 = {∅}) → 𝑥 = {∅})
2	1	ex 115	. . . 4 ⊢ (∃*𝑟 𝑟 TAp 2o → (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
3		df-stab 839	. . . 4 ⊢ (STAB 𝑥 = {∅} ↔ (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
4	2, 3	sylibr 134	. . 3 ⊢ (∃*𝑟 𝑟 TAp 2o → STAB 𝑥 = {∅})
5	4	adantr 276	. 2 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
6	5	exmid1stab 4304	1 ⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 838   = wceq 1398  ∃*wmo 2080   ⊆ wss 3201  ∅c0 3496  {csn 3673  EXMIDwem 4290  2_oc2o 6619   TAp wtap 7511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-tr 4193  df-exmid 4291  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-1o 6625  df-2o 6626  df-pap 7510  df-tap 7512

This theorem is referenced by:  exmidmotap  7523